It is trivial that (REF ) implies (REF ).
After passing to a subsequence by [1]}, [2]}, with no loss of generality we can assume that the pointed measured Gromov–Hausdorff (pmGH) convergence holds:
\(\left(M^n_i, \mathsf {d}_{g_i}, x_i, \frac{\mathrm {vol}_{g_i}}{\mathrm {vol}_{g_i}B_1(x_i)} \right) \stackrel{\mathrm {pmGH}}{\longrightarrow } (X, \mathsf {d}, x, \mathfrak {m})\) 
